Interacting Vertex Operators in Euclidean AdS₂

This post outlines the article I am preparing for submission to JHEP, Interacting Vertex Operators in Euclidean AdS₂ (read the abstract). It extends the free-field analysis of my thesis to the interacting theory and settles a question the thesis left open.

Where the thesis left off

In the free theory, normal-ordered vertex operators $V_\beta = {:}e^{i\beta\phi}{:}$ in Euclidean Poincaré $\text{AdS}_2$ map cleanly to boundary primaries through the bulk-to-boundary dictionary. The interesting physics shows up once you add a Sine-Gordon-type bulk interaction and ask what happens to the connected boundary correlators.

Connected generating functional

Starting from the Gaussian scalar theory, the connected correlators are organized by a connected generating functional. For the two-point function this is schematically

$$\langle V_{\beta_1} V_{\beta_2} \rangle_c = e^{-\beta_1\beta_2\, G_\Delta(\vec{x}_1, \vec{x}_2)} - 1,$$

with $G_\Delta$ the bulk propagator. A Euclidean Gell-Mann-Low expansion then generates the perturbative corrections from the interaction, order by order in the coupling $\lambda$.

Holographic renormalization

The first-order correction carries a logarithmic divergence from the internal AdS integration. The article does the holographic renormalization explicitly: a cutoff near the conformal boundary, a renormalization factor $Z(\epsilon)$ that absorbs the divergence, and a counterterm fixed so the renormalized operator

$$\mathcal{O}^{\text{ren}}_{\Delta_1}(p) = \frac{1}{i\beta}\lim_{\epsilon\to 0}\sqrt{Z(\epsilon)}\,\lim_{z\to 0} z^{-\Delta_1} V_\beta(\vec{x})\big|_{z=\epsilon}$$

is finite. The renormalization scale $M$ is arbitrary, and tracking it gives a beta function and the renormalized anomalous dimension $\Delta_1 = \Delta + \lambda\,\Delta^{(1)} + \mathcal{O}(\lambda^2)$. I work out the scheme dependence directly and compare the curved-space structure with the flat-space Sine-Gordon result.

The main result: no logarithmic CFT

The thesis hinted that the logarithmic terms and the apparent multiplet $\widetilde{\mathcal{O}} = \{\hat{\mathcal{O}}, {:}\mathcal{O}^2{:}\}$ might signal a logarithmic CFT on the boundary. Working through the connected correlators and the renormalization group more carefully, the article shows the opposite:

Despite the appearance of logarithmic terms, the resulting boundary theory shows ordinary anomalous scaling rather than logarithmic CFT structure.

In other words, once the renormalization is done consistently the logarithms reorganize into a conventional anomalous dimension. The operator algebra does not develop the non-diagonalizable mixing that defines a genuine logarithmic CFT.

Why it matters

This sharpens a recurring subtlety in holography: logarithms in AdS correlators are common, but they do not automatically imply logarithmic CFT structure. Pinning down exactly when interactions in curved backgrounds generate genuine logarithmic multiplets, and when they only renormalize scaling dimensions, is the direction I want to push further, toward higher-point functions and the operator product expansion.

The numerical side of this program is implemented in DiffQFT: differentiable Witten-diagram integration, neural surrogates, and a PINN for the Klein-Gordon equation.